To find the domain of the function $f(x) = \frac{4x + 1}{x^2 + x - 6}$, we need to determine the values of $x$ that make the function undefined. Since a rational function is undefined when its denominator equals zero, we begin by solving for the values of $x$ that make the denominator zero.

Step 1: Set the denominator equal to zero:

$x^2 + x - 6 = 0$

Step 2: Factor the quadratic equation:

$x^2 + x - 6 = (x - 2)(x + 3) = 0$

Step 3: Solve for $x$:

$x - 2 = 0 \quad \text{or} \quad x + 3 = 0$ $x = 2 \quad \text{or} \quad x = -3$

Step 4: Exclude these values from the domain:

Since the function is undefined when $x = 2$ and $x = -3$, the domain of the function is all real numbers except these two points.

Final Answer:

The domain of $f(x)$ is: $\boxed{(-\infty, -3) \cup (-3, 2) \cup (2, \infty)}$

Would you like more details or have any questions?

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Here are 5 related questions for further exploration:

1. How would the domain change if the numerator had a square root?
2. What are the vertical asymptotes of the function $f(x) = \frac{4x + 1}{x^2 + x - 6}$?
3. How do you find the horizontal asymptote of a rational function?
4. What would the graph of this function look like, and where would it have discontinuities?
5. Can you determine the behavior of the function near $x = -3$ and $x = 2$?

Tip: When factoring quadratics, always check if the coefficients suggest a simple factoring pattern (like two binomials).